Creation Functions

Creation of Structures

For any nonsquare integer \(D\) congruent to \(0\) or \(1\) modulo \(4\), binary quadratic forms of discriminant \(D\) may be created. The parent object of forms of discriminant \(D\) can be created using the following commands.

BinaryQuadraticForms(D): RngIntElt -> QuadBin

QuadraticForms(D): RngIntElt -> QuadBin

Create the structure of integral binary quadratic forms of discriminant \(D\).

Creation of Forms

Binary quadratic forms may be created by coercing a triple \([a,b,c]\) of integer coefficients into the parent structure of forms of discriminant \(D = b^2-4ac\). Other constructors are provided for constructing the group identity, prime forms, or allowing the omission of third element \(c\) of the sequence.

Identity(Q): QuadBin -> QuadBinElt

Q ! 1: QuadBin, RngIntElt -> QuadBinElt

Create the principal form in the structure \(Q\) of binary quadratic forms of discriminant \(D\). The principal form is a reduced form equivalent to \(X^2-D/4Y^2\) when \(D \equiv 0 \bmod 4,\) or \(X^2+XY-(D-1)/4Y^2\) when \(D \equiv 1 \bmod 4\).

Q ! [a, b, c]: QuadBin, RngIntElt, RngIntElt, RngIntElt -> QuadBinElt

elt< Q | a, b, c>: QuadBin, RngIntElt, RngIntElt, RngIntElt -> QuadBinElt

elt< Q | a, b>: QuadBin, RngIntElt, RngIntElt -> QuadBinElt

Returns the binary quadratic form \(aX^2+bXY+cY^2\) in the magma of forms \(Q\) of discriminant \(D\). Here \(c\) is determined by the solution of the equality \(D = b^2-4ac\); if no integer \(c\) exists satisfying this, an error will occur.

PrimeForm(Q, p): QuadBin, RngIntElt -> QuadBinElt

If \(p\) is a split prime or a ramified prime not dividing the conductor of the magma of quadratic forms \(Q\), returns a quadratic form \(pX^2+bXY+cY^2\) in \(Q\).